Halachah 1 – Calculating the First Longitude
All the previous calculations were preparation for determining whether the new moon can actually be seen. To begin this final process, one calculates the true position of the sun, the true position of the moon, and the position of the lunar node at the time of visibility. Then one subtracts the sun’s true position from the moon’s true position. The remainder is called the first longitude.
Halachah 2 – Calculating the First Latitude
After finding the moon’s true position and the position of the node, one calculates the moon’s latitude and determines whether it is northern or southern. This result is called the first latitude. The Rambam emphasizes that both the first longitude and first latitude must be kept ready, because the later visibility calculations depend on them.
Halachah 3 – When the First Longitude Clearly Prevents Visibility
If the first longitude is exactly 9 degrees or less, the moon definitely cannot be seen anywhere in Eretz Yisrael that night. In such a case, no further calculation is needed.
If the first longitude is greater than 15 degrees, the moon will certainly be visible throughout Eretz Yisrael, and no further calculation is needed.
If the first longitude is between 9 and 15 degrees, more detailed visibility calculations are required.
Halachah 4 – Different Thresholds Based on the Moon’s Zodiac Position
The previous rule applies when the moon is located from the beginning of Capricorn through the end of Gemini. However, if the moon is located from the beginning of Cancer through the end of Sagittarius, the thresholds change.
In that case:
- If the first longitude is 10 degrees or less, the moon will certainly not be visible.
- If the first longitude is greater than 24 degrees, the moon will certainly be visible.
- If it is between 10 and 24 degrees, further visibility calculations are needed.
Halachah 5 – Beginning the Visibility Calculations
The Rambam now begins the detailed calculations used when the simple first-longitude test is not enough. First, one checks which zodiac sign the moon is in. Based on that sign, one subtracts a certain amount from the first longitude. This corrected value is called the second longitude.
Halachah 6 – Why the Second Longitude Is Needed
The moon’s true astronomical position is not exactly the same as the place where it appears to the eye. This apparent displacement is called shinui hamareh, the “change of appearance,” or parallax. For longitude, this correction is always subtracted from the first longitude.
Halachah 7 – Correcting the First Latitude
The moon’s apparent latitude must also be corrected. If the moon’s first latitude is northern, the correction is subtracted from it. If the moon’s first latitude is southern, the correction is added to it. The result is called the second latitude.
Halachah 8 – The Latitude Correction by Zodiac Sign
The amount added or subtracted from the first latitude depends on the moon’s zodiac sign. For example:
- Aries: 9 parts
- Taurus: 10 parts
- Gemini: 16 parts
- Cancer: 27 parts
- Leo: 38 parts
- Virgo: 44 parts
- Libra: 46 parts
- Scorpio: 45 parts
- Sagittarius: 44 parts
- Capricorn: 36 parts
- Aquarius: 27 parts
- Pisces: 12 parts
These corrections account for how the moon appears from the observer’s location.
Halachah 9 – Preparing the Second Latitude
After applying the proper correction to the first latitude, one obtains the second latitude. One must know whether it is northern or southern and how many degrees and parts it contains, because the next stages depend on this value.
Halachah 10 – Calculating the Moon’s Inclination Correction
The next step is to take a fraction of the second latitude because the moon’s path is slightly curved or inclined. The fraction taken depends on where the moon is located in the zodiac. In some regions one takes two-fifths, in others one-third, one-fourth, one-fifth, one-sixth, half of a sixth, or a quarter of a sixth. In some zodiac regions no correction is taken at all. This fraction is called the moon’s circle correction.
Halachah 11 – Producing the Third Longitude
The moon’s circle correction is then applied to the second longitude.
If the moon is from Capricorn through Gemini:
- Northern latitude means subtract the moon’s circle correction.
- Southern latitude means add it.
If the moon is from Cancer through Sagittarius:
- Northern latitude means add it.
- Southern latitude means subtract it.
The result is called the third longitude. If no circle correction applies, the second longitude remains unchanged and becomes the third longitude.
Halachah 12 – Producing the Fourth Longitude and the Arc of Visibility
The third longitude is then adjusted according to the zodiac region in which it falls. Sometimes one adds a fraction of it, sometimes subtracts a fraction, and sometimes leaves it unchanged. The result is the fourth longitude.
Then one returns to the first latitude and takes two-thirds of it. This is called the city-height correction. If the moon’s latitude is northern, this correction is added to the fourth longitude. If the moon’s latitude is southern, it is subtracted. The final result is called the arc of visibility.
Halachah 13 – A Full Example of the Visibility Calculation
The Rambam gives a full example for the beginning of Thursday night, the second of Iyar in the reference year. The sun’s true position is calculated as 7 degrees and 9 parts in Taurus, the moon’s true position as 18 degrees and 36 parts in Taurus, and the moon’s first latitude as 3 degrees and 53 parts south. Subtracting the sun from the moon gives a first longitude of 11 degrees and 27 parts.
Since the moon is in Taurus, one subtracts 1 degree for the longitude correction, producing a second longitude of 10 degrees and 27 parts. Since the latitude is southern, one adds the latitude correction of 10 parts, producing a second latitude of 4 degrees and 3 parts. Because the moon is 18 degrees into Taurus, one takes one-fourth of that second latitude as the moon’s circle correction, which is about 1 degree and 1 part.
Halachah 14 – Completing the Example
Since the latitude is southern and the moon is in the relevant half of the zodiac, the moon’s circle correction is added to the second longitude, producing a third longitude of 11 degrees and 28 parts. Since this falls in Taurus, one adds one-fifth of it, producing a fourth longitude of 13 degrees and 46 parts.
Then one takes two-thirds of the first latitude, which gives the city-height correction of 2 degrees and 35 parts. Since the moon’s latitude is southern, this is subtracted from the fourth longitude. The final result is an arc of visibility of 11 degrees and 11 parts.
This same process is used to calculate the arc of visibility for any desired night.
Halachah 15 – Clear Rules Based on the Arc of Visibility
Once the arc of visibility has been calculated:
- If it is 9 degrees or less, the moon cannot be seen anywhere in Eretz Yisrael.
- If it is greater than 14 degrees, the moon will certainly be visible throughout Eretz Yisrael.
Halachah 16 – Borderline Cases Require Comparison
If the arc of visibility is between the beginning of the tenth degree and the end of the fourteenth degree, one must compare it to the first longitude. These comparison rules are called the visibility limits, or kitzei hare’iyah.
Halachah 17 – Visibility Limit: Around 10 Degrees
If the arc of visibility is greater than 9 degrees up to 10 degrees, the moon will be visible only if the first longitude is 13 degrees or more. If either value is less than that, it will not be visible.
Halachah 18 – Visibility Limit: Around 11 Degrees
If the arc of visibility is greater than 10 degrees up to 11 degrees, the moon will be visible only if the first longitude is 12 degrees or more. If not, it will not be visible.
Halachah 19 – Visibility Limit: Around 12 Degrees
If the arc of visibility is greater than 11 degrees up to 12 degrees, the moon will be visible only if the first longitude is 11 degrees or more. If either the arc or the first longitude is below the required amount, it will not be visible.
Halachah 20 – Visibility Limit: Around 13 Degrees
If the arc of visibility is greater than 12 degrees up to 13 degrees, the moon will be visible only if the first longitude is 10 degrees or more. If not, it will not be visible.
Halachah 21 – Visibility Limit: Around 14 Degrees
If the arc of visibility is greater than 13 degrees up to 14 degrees, the moon will be visible only if the first longitude is 9 degrees or more. This completes the set of visibility limits.
Halachah 22 – Applying the Example to the Visibility Limits
In the example from Iyar, the arc of visibility was 11 degrees and 11 parts, and the first longitude was 11 degrees and 27 parts. Since the arc is slightly more than 11 degrees and the first longitude is also more than 11 degrees, the moon will certainly be visible that night according to the established limits.
Halachah 23 – The Complexity of Moon Visibility
The Rambam reflects on how many calculations, additions, and subtractions are needed to determine moon visibility. The moon’s motion is far more complex than the sun’s, which is why the Sages said that the sun “knows its setting,” while the moon does not. Sometimes the moon’s path requires lengthening the calculation, sometimes shortening it, and therefore the arc of visibility may vary greatly.
Halachah 24 – Why These Calculations Are Trustworthy
The reasons behind all these calculations belong to the science of astronomy and mathematical calculation. The Rambam notes that Greek scholars wrote many books on these matters, while the earlier Jewish works from the time of the prophets did not survive. Nevertheless, truth is accepted because of proof, not because of who wrote it. Once a matter is demonstrated through clear evidence, it may be relied upon whether it was authored by Jewish prophets or by non-Jewish scholars.